Book•
Lévy processes and infinitely divisible distributions
01 Jan 2013-
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.
Citations
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01 Jan 2014
TL;DR: In this article, Kloeden, P., Ombach, J., Cyganowski, S., Kostrikin, A. J., Reddy, J.A., Pokrovskii, A., Shafarevich, I.A.
Abstract: Algebra and Famous Inpossibilities Differential Systems Dumortier.: Qualitative Theory of Planar Jost, J.: Dynamical Systems. Examples of Complex Behaviour Jost, J.: Postmodern Analysis Jost, J.: Riemannian Geometry and Geometric Analysis Kac, V.; Cheung, P.: Quantum Calculus Kannan, R.; Krueger, C.K.: Advanced Analysis on the Real Line Kelly, P.; Matthews, G.: The NonEuclidean Hyperbolic Plane Kempf, G.: Complex Abelian Varieties and Theta Functions Kitchens, B. P.: Symbolic Dynamics Kloeden, P.; Ombach, J.; Cyganowski, S.: From Elementary Probability to Stochastic Differential Equations with MAPLE Kloeden, P. E.; Platen; E.; Schurz, H.: Numerical Solution of SDE Through Computer Experiments Kostrikin, A. I.: Introduction to Algebra Krasnoselskii, M.A.; Pokrovskii, A.V.: Systems with Hysteresis Kurzweil, H.; Stellmacher, B.: The Theory of Finite Groups. An Introduction Lang, S.: Introduction to Differentiable Manifolds Luecking, D.H., Rubel, L.A.: Complex Analysis. A Functional Analysis Approach Ma, Zhi-Ming; Roeckner, M.: Introduction to the Theory of (non-symmetric) Dirichlet Forms Mac Lane, S.; Moerdijk, I.: Sheaves in Geometry and Logic Marcus, D.A.: Number Fields Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis Matoušek, J.: Using the Borsuk-Ulam Theorem Matsuki, K.: Introduction to the Mori Program Mazzola, G.; Milmeister G.; Weissman J.: Comprehensive Mathematics for Computer Scientists 1 Mazzola, G.; Milmeister G.; Weissman J.: Comprehensive Mathematics for Computer Scientists 2 Mc Carthy, P. J.: Introduction to Arithmetical Functions McCrimmon, K.: A Taste of Jordan Algebras Meyer, R.M.: Essential Mathematics for Applied Field Meyer-Nieberg, P.: Banach Lattices Mikosch, T.: Non-Life Insurance Mathematics Mines, R.; Richman, F.; Ruitenburg, W.: A Course in Constructive Algebra Moise, E. E.: Introductory Problem Courses in Analysis and Topology Montesinos-Amilibia, J.M.: Classical Tessellations and Three Manifolds Morris, P.: Introduction to Game Theory Nikulin, V.V.; Shafarevich, I. R.: Geometries and Groups Oden, J. J.; Reddy, J. N.: Variational Methods in Theoretical Mechanics Øksendal, B.: Stochastic Differential Equations Øksendal, B.; Sulem, A.: Applied Stochastic Control of Jump Diffusions Poizat, B.: A Course in Model Theory Polster, B.: A Geometrical Picture Book Porter, J. R.; Woods, R.G.: Extensions and Absolutes of Hausdorff Spaces Radjavi, H.; Rosenthal, P.: Simultaneous Triangularization Ramsay, A.; Richtmeyer, R.D.: Introduction to Hyperbolic Geometry Rees, E.G.: Notes on Geometry Reisel, R. B.: Elementary Theory of Metric Spaces Rey, W. J. J.: Introduction to Robust and Quasi-Robust Statistical Methods Ribenboim, P.: Classical Theory of Algebraic Numbers Rickart, C. E.: Natural Function Algebras Roger G.: Analysis II Rotman, J. J.: Galois Theory Jost, J.: Compact Riemann Surfaces Applications ́ Introductory Lectures on Fluctuations of Levy Processes with Kyprianou, A. : Rautenberg, W.; A Concise Introduction to Mathematical Logic Samelson, H.: Notes on Lie Algebras Schiff, J. L.: Normal Families Sengupta, J.K.: Optimal Decisions under Uncertainty Séroul, R.: Programming for Mathematicians Seydel, R.: Tools for Computational Finance Shafarevich, I. R.: Discourses on Algebra Shapiro, J. H.: Composition Operators and Classical Function Theory Simonnet, M.: Measures and Probabilities Smith, K. E.; Kahanpää, L.; Kekäläinen, P.; Traves, W.: An Invitation to Algebraic Geometry Smith, K.T.: Power Series from a Computational Point of View Smoryński, C.: Logical Number Theory I. An Introduction Stichtenoth, H.: Algebraic Function Fields and Codes Stillwell, J.: Geometry of Surfaces Stroock, D.W.: An Introduction to the Theory of Large Deviations Sunder, V. S.: An Invitation to von Neumann Algebras Tamme, G.: Introduction to Étale Cohomology Tondeur, P.: Foliations on Riemannian Manifolds Toth, G.: Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems Wong, M.W.: Weyl Transforms Xambó-Descamps, S.: Block Error-Correcting Codes Zaanen, A.C.: Continuity, Integration and Fourier Theory Zhang, F.: Matrix Theory Zong, C.: Sphere Packings Zong, C.: Strange Phenomena in Convex and Discrete Geometry Zorich, V.A.: Mathematical Analysis I Zorich, V.A.: Mathematical Analysis II Rybakowski, K. P.: The Homotopy Index and Partial Differential Equations Sagan, H.: Space-Filling Curves Ruiz-Tolosa, J. R.; Castillo E.: From Vectors to Tensors Runde, V.: A Taste of Topology Rubel, L.A.: Entire and Meromorphic Functions Weintraub, S.H.: Galois Theory
401 citations
Cites background from "Lévy processes and infinitely divis..."
...There can be no doubt, particularly to the more experienced reader, that the current text has been heavily influenced by the outstanding books of Bertoin (1996) and Sato (1999), and especially the former which also takes a predominantly pathwise approach to its content....
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...See Zolotarev (1986), Sato (1999) and (Samorodnitsky and Taqqu, 1994) for further details of all the facts given in this paragraph....
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...The interested reader is referred to Lukacs (1970) or Sato (1999), to name but two of many possible references....
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TL;DR: In this article, several definitions of the Riesz fractional Laplace operator in R^d have been studied, including singular integrals, semigroups of operators, Bochner's subordination, and harmonic extensions.
Abstract: This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain.
372 citations
Cites background from "Lévy processes and infinitely divis..."
...Distributional definition of L is also studied in [7, 34, 42], see also [3, 43]....
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...Yet another way to show (10) involves vague convergence of tpt(z)dz to ν(z)dz = cd,α|z|dz as t → 0, which is a general result in the theory of convolution semigroups, see [43]....
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TL;DR: A closed formula for prices of perpetual American call options in terms of the overall supremum of the Lévy process, and a corresponding closed formulas for perpetual American put options involving the infimum of the after-mentioned process are obtained.
Abstract: Consider a model of a financial market with a stock driven by a Levy process and constant interest rate. A closed formula for prices of perpetual American call options in terms of the overall supremum of the Levy process, and a corresponding closed formula for perpetual American put options involving the infimum of the after-mentioned process are obtained. As a direct application of the previous results, a Black-Scholes type formula is given. Also as a consequence, simple explicit formulas for prices of call options are obtained for a Levy process with positive mixed-exponential and arbitrary negative jumps. In the case of put options, similar simple formulas are obtained under the condition of negative mixed-exponential and arbitrary positive jumps. Risk-neutral valuation is discussed and a simple jump-diffusion model is chosen to illustrate the results.
269 citations
TL;DR: For the isotropic unimodal probability convolutional semigroups, this article gave sharp bounds for their Levy-Khintchine exponent with Matuszewska indices strictly between 0 and 2.
172 citations
01 May 2013
TL;DR: In this paper, the authors review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics.
Abstract: We review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics. The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deterministic modeling of extreme events, via continuous and discrete dynamic models. The applications include climatic, seismic and socio-economic events, along with their prediction. Two important results refer to (i) the complementarity of spectral analysis of a time series in terms of the continuous and the discrete part of its power spectrum; and (ii) the need for coupled modeling of natural and socio-economic systems. Both these results have implications for the study and prediction of natural hazards and their human impacts.
166 citations
References
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TL;DR: In this paper, the authors provide a simple example of regulation risk, which is that, in certain situations, the prudential rules imposed by the regulator in the framework of the Basel II/III Accords or Solvency II directive are themselves the source of a systemic risk.
Abstract: In this note, we provide a simple example of regulation risk. The idea is that, in certain situations, the very prudential rules (or, rather, some of them) imposed by the regulator in the framework of the Basel II/III Accords or Solvency II directive are themselves the source of a systemic risk. The instance of regulation risk that we bring to light in this work can be summarised as follows: wrongly assuming that prices evolve in a continuous fashion when they may in fact display large negative jumps, and trying to minimise Value at Risk (VaR) under a constraint of minimal volume of activity leads in effect to behaviours that will maximise VaR. Although much stylised, our analysis highlights some pitfalls of model-based regulation.
5 citations
Dissertation•
22 May 2013
TL;DR: In this paper, the authors consider the problem of European-style derivatives pricing in geometric Levy models, where the price process of the underlying follows a process with independent and stationary increments.
Abstract: This thesis deals with two basic problems of Mathematical Finance, namely the pricing and hedging of European-style derivatives. We analyze these questions in models that describe the asset underlying the derivative by a process with jumps and stochastic volatility. However, we are not interested in exact solutions to the mentioned problems but in reasonable approximations that allow for a better insight into the structure of the respective question.
More precisely, we consider hedging problems in geometric Levy models, i.e., in models where the logarithmic price process of the underlying follows a process with independent and stationary increments. In this kind of models, there typically exist no perfect hedging strategies. We quantify the remaining risk of a self-financing trading strategy by its mean squared hedging error, i.e., the second moment of the difference between the payoff of the derivative and the terminal wealth of the hedging portfolio.
We study the question of derivative pricing in a comprehensive model class that encompasses geometric Levy models and several stochastic volatility models from the literature. In doing so, we consider prices that are compatible with the absence of arbitrage, i.e., prices that do not allow for riskless gains.
For several hedging strategies, for their hedging errors, as well as for derivative prices in the described framework, the literature provides semi-explicit representations that can be efficiently evaluated numerically for many parametric models. However, these representations admit little understanding, e.g., of the determining factors of the respective quantity. We develop approximate solutions that provide more insight in this respect. To this end, we interpret the complex model with jumps and stochastic volatility as a perturbed Black-Scholes model, and we compute correction terms of second order. Our approach differs from traditional perturbation techniques in the sense that in our case, there is no univariate problem-inherent parameter that quantifies the amount of perturbation. Therefore, we develop a general framework for perturbation approaches in this situation, and we apply this approach in the models under consideration.
The approximate solutions obtained in this way consist of few moments of components of the asset price process as well as of sensitivities (greeks) of the Black-Scholes derivative price. In particular, the formulas do not depend on the fine structure of the considered model and are robust in this sense. We show in detailed numerical experiments that our approximations yield satisfactory results in several parametric models from the literature.
5 citations
TL;DR: In this article, it was shown that the Pareto law is the only possible weak limit for the convergence of a stochastic process near t = 0 of a decreasing function.
Abstract: We prove several results on the behavior near t=0 of $Y_t^{-t}$ for certain $(0,\infty)$-valued stochastic processes $(Y_t)_{t>0}$. In particular, we show for L\'{e}vy subordinators that the Pareto law on $[1,\infty)$ is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of $tL(Y_t)$ as $t\to0$ for a decreasing function $L$ that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.
5 citations
Posted Content•
TL;DR: An explicit new distributional identity is obtained for the case where the L\'evy process is an independent sum of a Brownian motion and a general subordinator in terms of a geometrically distributed sum of independent random variables.
Abstract: The goal is to identify the class of distributions to which the distribution of the maximum of a L\'evy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the L\'evy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing L\'evy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczeck-Khinchine formula for stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.
5 citations
Dissertation•
15 May 2013
TL;DR: This dissertation recalls the pair-copula construction for random variables, presents different extensions and applies the concept in an empirical study, and introduces the new pair-Levy copula construction to model the jump dependence of high-dimensional Levy processes.
Abstract: Modeling high-dimensional dependence structures with parametric dependence models is a challenging and important part in statistics. The dependence models that we discuss here are based on bivariate building blocks and we illustrate how to assemble these building blocks to multivariate dependence structures. In the first part of the dissertation, we recall the pair-copula construction for random variables, present different extensions and apply the concept in an empirical study. In the second part, we introduce the new pair-Levy copula construction to model the jump dependence of high-dimensional Levy processes. Moreover, we present simulation and estimation algorithms, conduct a simulation study and discuss various applications of this concept.
5 citations